The first-order optimality conditions for the optimal \(\mathcal {H}_2\)  model reduction was given in [1]}, which is known as Wilson conditions. Wilson conditions are based on the first derivatives of (REF ) with respect to \(\hat{A}\) , \(\hat{B}\)  and \(\hat{C}\)  as follows:
\(\nabla \Xi _{\hat{A}}= 2(\hat{Q}\hat{P}+W^TEV),\,\nabla \Xi _{\hat{B}}= 2(\hat{Q}\hat{B}+W^TE^{-1}B), \,\nabla \Xi _{\hat{C}}= 2(\hat{C}\hat{P}-CV).\) 
